29,968 research outputs found
Approximation of fuzzy numbers by convolution method
In this paper we consider how to use the convolution method to construct
approximations, which consist of fuzzy numbers sequences with good properties,
for a general fuzzy number. It shows that this convolution method can generate
differentiable approximations in finite steps for fuzzy numbers which have
finite non-differentiable points. In the previous work, this convolution method
only can be used to construct differentiable approximations for continuous
fuzzy numbers whose possible non-differentiable points are the two endpoints of
1-cut. The constructing of smoothers is a key step in the construction process
of approximations. It further points out that, if appropriately choose the
smoothers, then one can use the convolution method to provide approximations
which are differentiable, Lipschitz and preserve the core at the same time.Comment: Submitted to Fuzzy Sets and System at Sep 18 201
Review of modern numerical methods for a simple vanilla option pricing problem
Option pricing is a very attractive issue of financial engineering and optimization. The problem of determining the fair price of an option arises from the assumptions made under a given financial market model. The increasing complexity of these market assumptions contributes to the popularity of the numerical treatment of option valuation. Therefore, the pricing and hedging of plain vanilla options under the Black–Scholes model usually serve as a bench-mark for the development of new numerical pricing approaches and methods designed for advanced option pricing models. The objective of the paper is to present and compare the methodological concepts for the valuation of simple vanilla options using the relatively modern numerical techniques in this issue which arise from the discontinuous Galerkin method, the wavelet approach and the fuzzy transform technique. A theoretical comparison is accompanied by an empirical study based on the numerical verification of simple vanilla option prices. The resulting numerical schemes represent a particularly effective option pricing tool that enables some features of options that are depend-ent on the discretization of the computational domain as well as the order of the polynomial approximation to be captured better
Analysis of the Performance Measures of a Non-Markovian Fuzzy Queue via Fuzzy Laplace Transforms Method
Laplace transforms play an essential role in the analysis of classical non-Markovian queueing systems. The problem addressed here is whether the Laplace transform approach is still valid for determining the characteristics of such a system in a fuzzy environment. In this paper, fuzzy Laplace transforms are applied to analyze the performance measures of a non-Markovian fuzzy queueing system FM/ FG/1. Starting from the fuzzy Laplace transform of the service time distribution, we define the fuzzy Laplace transform of the distribution of the dwell time of a customer in the system. By applying the properties of the moments of this distribution, the derivative of this fuzzy transform makes it possible to obtain a fuzzy expression of the average duration of stay of a customer in the system. This expression is the fuzzy formula of the same performance measure that can be obtained from its classical formula by the Zadeh extension principle. The fuzzy queue FM/ FE_k /1 is particularly treated in this text as a concrete case through its service time distribution. In addition to the fuzzy arithmetic of L-R type fuzzy numbers, based on the secant approximation, the properties of the moments of a random variable and Little's formula are used to compute the different performance measures of the system. A numerical example was successfully processed to validate this approach. The results obtained show that the modal values of the performance measures of a non-Markovian fuzzy queueing system are equal to the performance measures of the corresponding classical model computable by the Pollaczeck-Khintchine method. The fuzzy Laplace transforms approach is therefore applicable in the analysis of a fuzzy FM/FG/1 queueing system in the same way as the classical M/G/1 model
Weighted Banzhaf power and interaction indexes through weighted approximations of games
The Banzhaf power index was introduced in cooperative game theory to measure
the real power of players in a game. The Banzhaf interaction index was then
proposed to measure the interaction degree inside coalitions of players. It was
shown that the power and interaction indexes can be obtained as solutions of a
standard least squares approximation problem for pseudo-Boolean functions.
Considering certain weighted versions of this approximation problem, we define
a class of weighted interaction indexes that generalize the Banzhaf interaction
index. We show that these indexes define a subclass of the family of
probabilistic interaction indexes and study their most important properties.
Finally, we give an interpretation of the Banzhaf and Shapley interaction
indexes as centers of mass of this subclass of interaction indexes
Hierarchical Multi-resolution Mesh Networks for Brain Decoding
We propose a new framework, called Hierarchical Multi-resolution Mesh
Networks (HMMNs), which establishes a set of brain networks at multiple time
resolutions of fMRI signal to represent the underlying cognitive process. The
suggested framework, first, decomposes the fMRI signal into various frequency
subbands using wavelet transforms. Then, a brain network, called mesh network,
is formed at each subband by ensembling a set of local meshes. The locality
around each anatomic region is defined with respect to a neighborhood system
based on functional connectivity. The arc weights of a mesh are estimated by
ridge regression formed among the average region time series. In the final
step, the adjacency matrices of mesh networks obtained at different subbands
are ensembled for brain decoding under a hierarchical learning architecture,
called, fuzzy stacked generalization (FSG). Our results on Human Connectome
Project task-fMRI dataset reflect that the suggested HMMN model can
successfully discriminate tasks by extracting complementary information
obtained from mesh arc weights of multiple subbands. We study the topological
properties of the mesh networks at different resolutions using the network
measures, namely, node degree, node strength, betweenness centrality and global
efficiency; and investigate the connectivity of anatomic regions, during a
cognitive task. We observe significant variations among the network topologies
obtained for different subbands. We, also, analyze the diversity properties of
classifier ensemble, trained by the mesh networks in multiple subbands and
observe that the classifiers in the ensemble collaborate with each other to
fuse the complementary information freed at each subband. We conclude that the
fMRI data, recorded during a cognitive task, embed diverse information across
the anatomic regions at each resolution.Comment: 18 page
Measuring the interactions among variables of functions over the unit hypercube
By considering a least squares approximation of a given square integrable
function by a multilinear polynomial of a specified
degree, we define an index which measures the overall interaction among
variables of . This definition extends the concept of Banzhaf interaction
index introduced in cooperative game theory. Our approach is partly inspired
from multilinear regression analysis, where interactions among the independent
variables are taken into consideration. We show that this interaction index has
appealing properties which naturally generalize the properties of the Banzhaf
interaction index. In particular, we interpret this index as an expected value
of the difference quotients of or, under certain natural conditions on ,
as an expected value of the derivatives of . These interpretations show a
strong analogy between the introduced interaction index and the overall
importance index defined by Grabisch and Labreuche [7]. Finally, we discuss a
few applications of the interaction index
Spectral geometry with a cut-off: topological and metric aspects
Inspired by regularization in quantum field theory, we study topological and
metric properties of spaces in which a cut-off is introduced. We work in the
framework of noncommutative geometry, and focus on Connes distance associated
to a spectral triple (A, H, D). A high momentum (short distance) cut-off is
implemented by the action of a projection P on the Dirac operator D and/or on
the algebra A. This action induces two new distances. We individuate conditions
making them equivalent to the original distance. We also study the
Gromov-Hausdorff limit of the set of truncated states, first for compact
quantum metric spaces in the sense of Rieffel, then for arbitrary spectral
triples. To this aim, we introduce a notion of "state with finite moment of
order 1" for noncommutative algebras. We then focus on the commutative case,
and show that the cut-off induces a minimal length between points, which is
infinite if P has finite rank. When P is a spectral projection of , we work
out an approximation of points by non-pure states that are at finite distance
from each other. On the circle, such approximations are given by Fejer
probability distributions. Finally we apply the results to Moyal plane and the
fuzzy sphere, obtained as Berezin quantization of the plane and the sphere
respectively.Comment: Reference added. Minor corrections. Published version. 38 pages, 2
figures. Journal of Geometry and Physics 201
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